1. Field of the Invention
The invention relates to color management in the context of printing. More specifically, the invention relates methods for characterizing the color mixing behavior of printers.
2. Description of the Related Art
Separation of Color into a Colorant Set
Color images are typically captured and stored in a colorimetric color space, such as for example an RGB color space or more specifically an sRGB color space.
With color space is meant a space that represents a number of quantities of an object that characterize its color. In most practical situations, colors will be represented in a 3-dimensional space such as the CIE XYZ space, CIELAB or CIECAM02. However, also other characteristics can be used such as multispectral values based on filters that are not necessarily a linear transformation of the color matching functions. The values represented in a color space are referred to as color values. Color spaces are also referred to as device independent spaces.
With colorant space is meant an n-dimensional space with n the number of independent variables with which the color device can be addressed. In the case of an offset printing press the dimension of the colorant space corresponds to the number of inks of the press. As normally CMYK inks are used, the dimension of the colorant space is four. Colorant spaces are also referred to as device dependent spaces.
The colorant domain is defined by all possible combinations of colorant values, ranging from 0% to 100%. If there are no extra colorant limitations, the colorant domain is an n-dimensional cube, defined by the boundary planes with one ink equal to 0 or 100%. However, in most cases also one or multiple ink limitations have to be taken into account as a number of colorant combinations are not acceptable to be printed. Hence the colorant domain is reduced by these ink limitations. Most commonly these ink limitations are expressed as a set of linear inequalities that constraint the colorant domain. In general the colorant domain of the n-dimensional colorant space is defined by 2·n+s hyperplanes; i.e. 2·n boundary planes plus a number of s linear ink limitations. These 2·n+s hyperplanes will be called colorant boundary planes. These planes will be defined such that points of the colorant domain are at the negative or zero side for all colorant boundary planes. If a colorant is contained by one or multiple colorant boundary planes and is at the negative side for all the other colorant boundary planes, the colorant lies at the boundary of the colorant domain. If the colorant is at the negative side for all colorant boundary planes, the colorant is inside the colorant domain.
The color gamut is defined as the set of colors obtained by printing all colorant combinations from the colorant domain.
Most color printers use a set of CMYK colorants for printing that is sometimes complemented with orange/red, green, blue/violet, light cyan, light magenta and/or light gray colorants.
The conversion of a color into a set of colorants for the purpose of its rendering on a printer is called “color separation” or simply “separation”. It is recognized that this is a technically complex problem for a number of reasons.
In the first place does the conversion of a color into a colorant set generally not yield a unique solution: multiple colorant sets can be used to render a specific single color. Mathematically speaking the solution of the transformation of a color into a set of four or more colorants is said to be “undetermined”. This state of being indeterminate can be resolved by imposing additional constraints on the range of otherwise available solutions. In order to avoid technical problems during the printing, the imposed constraints should be such that they preserve continuity, i.e. they should guarantee that any continuous path of colors in the color space is transformed into a range of colorant sets that form a continuous path in the colorant space.
A second problem is that the printable color gamut of a printer is never congruent with the color gamut of an image. In order to preserve the proper rendering of gradients and to avoid clipping, the color gamut of an image needs to be perceptually mapped onto the printable color gamut of a printer.
Printer Models
In the following we will consider subsets of the colorant domain, in which one of the colorants takes on different values, ranging from 0% to 100%, while the other colorants have a value of 0%. We will call these sets 1-ink subprocesses. In the same way colorant combinations where only m out of n inks are varying form an m-ink subprocess.
If however, one colorant of an n-dimensional colorant space is varying from 0 till 100%, and the remaining colorants are constant, and not all are equal to 0%, the subset is called a 1-ink process. If the domain of the 1-ink process lies at the boundary of the colorant domain, the 1-ink process is referred to as a 1-ink boundary process. In analogy m-ink processes and m-ink boundary processes can be defined.
An edge point p of the colorant domain is a point belonging to the colorant domain and for which, for every direction u in the colorant space, we can construct a line segment l with mid p and direction u, such that l is not entirely contained in the colorant domain. With no ink limitations present, these are just the colorant combinations where all values are either 0% or 100%. When ink limitations are present the set of edge points contains all the edge points of the unlimited colorant domain that lie within the ink limitation. Additional edge points are obtained as the intersection of the ink limitation plane with the colorant domain boundary.
In general an edge point in an n-dimensional colorant space is obtained by the intersection of n non-parallel colorant boundary planes; i.e. there are no 2 colorant boundary planes which are parallel.
In analogy with edge points, 1-dimensional edge processes can be defined by the intersection of n−1 non-parallel colorant boundary planes. And in general, p-dimensional edge processes are obtained by the intersection of n-p non-parallel colorant boundary planes.
Most color separation technologies are based on the use of a mathematical “printer model”. Such a printer model predicts what color results if a set of colorants are printed on a specific substrate. Since a printer model acts “forward”, i.e. predicts color as a function of amounts of colorants, it needs to be “inverted” for the purpose of color separation. This process is called “printer model inversion” or simply “model inversion”.
Because model inversion tends to be a computationally intensive process, it is usually done only for a (rectangular) sampling of colors in the color space that form the entries of a look-up-table. The result of the model inversion for each entry is stored in the data points of the look-up-table and a fast linear interpolation technique is used for obtaining colorant sets for colors in between the entries.
A well-known printer model is provided by the “Neugebauer equations”. These polynomials predict a color as an additive mixture the colors of the substrate, the primary colorants and of the various overlapping combinations in proportions that depend on the halftone dot percentages of the colorants. It is implicitly assumed in this model that the relative positions of the halftone dots are randomized. The coefficients of the polynomials are obtained by printing a “printer characterization target” that comprises a sufficient set of patches with known colorant combinations and measuring the color of these patches. By replacing in the Neugebauer polynomials the known colorant values of the printer target and the measured color values, a set of linear equations is obtained from which the unknown coefficients can be calculated using linear algebra.
Several variations of the original Neugebauer equations exist. In one variation (“spectral Neugebauer equations”) the simple additive color mixing is replaced by the addition of spectral components. In yet another variation (“Neugebauer equations with localized coefficients”), the colorant space is subdivided into a set of contiguous rectangular sub-regions that each have their own set of coefficients. In yet another variation, each rectangular sub-region is further subdivided into a contiguous set of tetrahedrons for which a set of coefficients is obtained. The Neugebauer equations in that case degenerate to first order (“linear”) polynomials.
More information on the Neugebauer equations and a technique for their inversion is found in the publication: Marc MAHY, et al. Inversion of the Neugebauer equations. Color Research. 6 Dec. 1996, vol. 21, no. 6, p. 404-411.
A well-known model that predicts light absorption and color as a function of colorant concentrations c inside a transmissive medium having a thickness X is the Beer-Lambert law. In FIG. 1 a light beam having an intensity I0 passes through a medium 100 having a thickness X and containing a colorant concentration c. The colorant reduces the intensity I0 to a value I. The value x refers to the distance in the medium measured from the top layer in a direction that is perpendicular to medium.
A very thin layer 101 having a thickness dx absorbs or scatters an amount of light di(x) that is proportional with the extinction coefficient ε of the colorant, the concentration c of the colorant and the intensity of the i(x) of the light beam at the position x:
                                          di            ⁡                          (              x              )                                x                =                  -                      c            .            ɛ            .                          i              ⁡                              (                x                )                                                                        (        1        )            
Integrating the above differential equation over the distance X leads to the expression that is known as the Beer-Lambert law:
                              I                      I            0                          =                  e                      -                          ɛ              .              c              .              X                                                          (        2        )            
By applying Beer-Lambert law on the spectral components of an incident light beam, it is possible to predict not only how the intensity of the light beam changes, but also its spectral shape and the corresponding color.
                                          I            ⁡                          (              λ              )                                                          I              0                        ⁡                          (              λ              )                                      =                  e                      -                                          ɛ                ⁡                                  (                  λ                  )                                            .              c              .              X                                                          (        3        )            
More information on the Beer-Lambert law is found in WYSZECKI, Günter, et al. Color Science: Concepts and Methods, Quantitative Data and Formulae. New York: John Wiley, 1982. ISBN 0471021067. p. 30-31.
If besides absorption also back scattering is to be modeled, the Kubelka-Munk model can be used. FIG. 2 shows a colorant layer 200 that is printed on an opaque background 201. The reflectance of the colorant layer is R, and the reflectance of the background layer is Rg. The colorant layer has a thickness X. Distances x are measured from the top of the background layer towards the top of the colorant layer.
In the Kubelka-Munk model a thin layer 202 can be identified having a thickness dx through which two light fluxes pass: a first flux i(x) that follows the direction of the component of an incident light beam that is perpendicular to the surface of the colorant layer and a second flux j(x) that follows the opposite direction.
Both light fluxes undergo attenuation due to absorption and back scattering. The quantity K denotes an absorption coefficient of a flux going through a thin layer having a thickness dx, i.e. the amount of flux that is lost due absorption. The quantity S denotes a backward scattering coefficient of a flux going by a thin layer having a thickness dx, i.e. the amount of the flux that is lost because it is reflected backwards. In a thin layer having a thickness dx at a position x the following differential equations are devised that describe the attenuation of both the flux i(x) and j(x):
                                          di            ⁡                          (              x              )                                dx                =                                            (                              K                +                S                            )                        .                          i              ⁡                              (                x                )                                              -                      S            .                          j              ⁡                              (                x                )                                                                        (        4        )                                                      dj            ⁡                          (              x              )                                dx                =                              -                                          (                                  K                  +                  S                                )                            .                              j                ⁡                                  (                  x                  )                                                              -                      S            .                          i              ⁡                              (                x                )                                                                        (        5        )            
Integrating this set of differential equations leads to the following expressions:
                    R        =                              1            -                                          R                g                            ⁡                              (                                  a                  -                                      b                    .                                          ctgh                      ⁡                                              (                                                  b                          .                          S                          .                          X                                                )                                                                                            )                                                          a            -                          R              g                        +                          b              .                              ctgh                ⁡                                  (                                      b                    .                    S                    .                    X                                    )                                                                                        (        6        )            wherein:
                    a        =                                            S              +              K                        S                    ⁢                                          ⁢          and                                    (        7        )                                b        =                              (                                          a                2                            -              1                        )                                              (        8        )            “ctgh” refers to “cotangent hyperbolicus”. The above equations are valid to calculate the reflectance of a colorant for the case of a narrow-band spectrum. By using the equations for the spectral components of the colorant layer and the background layer, it is possible to use the Kubelka-Munk model for calculating the spectrum of the reflected light beam and correspondingly its color. In that case the values for absorption coefficient K(λ) and the scattering coefficient S(λ) become dependent on the wavelength λ.
More information on the Kubelka-Munk model is found in WYSZECKI, Günter, et al. Color Science: Concepts and Methods, Quantitative Data and Formulae. New York: John Wiley, 1982. ISBN 0471021067. p. 221-222 and 785-786.
Dot Area Gain
Dot area gain in the printing industry refers to the effect that the area of halftone dots in a printed image appears to be larger than the size of the corresponding halftone dots in the digital raster image. The main cause of this is a combination of physical ink spread that occurs during the printing and optical internal light reflection near the halftone dot boundaries in the substrate. Both effects cause that the printed halftone dots absorb more light, i.e. have a larger apparent dot area than is expected based on the area of the digital halftone dots.
If the digital dot area of a halftone is denoted as “a”, the apparent dot area can be denoted as “a′”. The dot gain “g” itself is then defined as g=a′−a.
The apparent dot area a′ can be calculated by means of the Murray Davis equation:
                              a          ′                =                              1            -                          10                              (                                                      -                                          D                      t                                                        -                                      D                    p                                                  )                                                          1            -                          10                              (                                                      -                                          D                      s                                                        -                                      D                    p                                                  )                                                                        (        9        )            in which: Dt refers to a measured density of a halftone tint, Dp refers to a measured density of the paper white and Ds refers to the density of a solid color (a=100%) printed with the same ink as the halftone tint. In the above formula the densities are absolute densities.
When densities Dt are measured relative to the paper-white the value of Dp becomes equal to zero and the above formula is simplified to:
                              a          ′                =                              1            -                          10                              (                                  -                                      D                    t                                                  )                                                          1            -                          10                              (                                  -                                      D                    s                                                  )                                                                        (        10        )            
For halftones printed with black ink a visual filter is used, for halftones printed with cyan ink a red filter, for halftones printed with magenta a green filter and for halftones printed with yellow ink a blue filter is used.
By plotting the apparent dot area a′ as a function of the digital dot area a, a dot area curve is obtained, similar to the one shown in FIG. 4.
More information on the subject of dot area gain and the use of the Murray Davis equation is found in the publication SHARMA GAURAV, “Digital Color Imaging Handbook”, CRC Press LLC, 2003, ISBN 0-8493-0900-X., pages 220-222.
The Murray Davis model was originally designed to describe the effect of a change of halftone dot area during printing on the density of the printed tint. In many printing processes, however, the amount of ink is not only (or even not at all) controlled by modulating a dot area, but also by changing the thickness of a printed ink or its density. In such cases it makes little sense to use the terms “digital dot area”, “apparent dot area” or “dot area gain”. Instead the more general terms: “digital tone value”, “measured tone value” and “tone value increase” are preferable.
Regularization of a Printer Target
In many cases the color data that is obtained from measuring a printer target cannot be used without a prior editing step that is also called a “regularization” step. There are several reasons that justify the need for a prior regularization.
A first reason is that many printer characterization targets comprise patches that suffer from physical defects such as scratches, stripes and banding (in the case of inkjet printing), uneven toning (in the case of toner based printing systems).
A second reason is related to gloss effects such as differences in specular reflection.
A third reason are the complex physical and optical effects during printing that result in unexpected behavior such as additivity failure, whereby a higher amount of colorant introduces an unexpected decrease in absorption, rather than the opposite as in a “well behaving” printing process. Such additively failure results in the three-dimensional equivalent of non-monotonicity in a one-dimensional print process, making model inversion much more difficult.
Such irregularities, when not properly dealt with, can result in discontinuities in color separations when smooth gradations are rendered.
A second problem is that irregularities in the printer characterization target translate in an irregular shape of the printable gamut when it is calculated. Such an irregular shape greatly complicates the process of mapping the color gamut of an image that is to be printed onto the color gamut of the printing process.
A solution to deal with these problems is provided in the publication WO 2013124369 A (AGFA GRAPHICS NV) 29 Aug. 2013. [46a] The U.S. Pat. No. 6,654,143, having a priority date of 1999 Oct. 28 and assigned to Xerox Corporation, teaches a method for using a printer target with a reduced number of color patches for characterizing a printer. A first (complete) printer characterization target is printed and measured (32). From the measurement a 16 primary Neugebauer model printer model is derived, with corrections for the measured optical (34) and physical (36) dot gain. The colors predicted by the model are compared with the measured colors for obtaining correction factors (44). When a different medium is used, a second (reduced) printer characterization target is printed and measured (46) that only comprises the 16 Neugebauer primaries. This model in combination with the corrections for optical (34) and physical (36) dot gain derived from the first model and the correction factors (44) is used for accurately predicting the color of a cmyk ink combination on the new substrate. Further embodiments disclose the use of the Kubelka-Munk model for predicting optical dot gain and the use of the spectral Neugebauer equations to improve the precision of colors predicted by the Neugebauer equations.
The U.S. Pat. No. 8,654,395, having a priority date of 2010 Feb. 12 and assigned to Heidelberger Druckmaschinen AG, discloses the use of a reduced printer target that is spectrally measured in combination with spectral models for accurately predicting color. A first type of model accurately predicts for each spectral band the tone value increase. A second type of model accurately predicts for each spectral band the color mixing behavior for various overprints of the inks. According to a preferred embodiment the segmented spectral Yule Nielsen Neugebauer model is used for the latter purpose.